The correct answer is $83$. It's given that in the figure, $AC=CD$. Thus, triangle $ACD$ is an isosceles triangle and the measure of angle $CDA$ is equal to the measure of angle $CAD$. The sum of the measures of the interior angles of a triangle is $180°$. Thus, the sum of the measures of the interior angles of triangle $ACD$ is $180°$. It's given that the measure of angle $ACD$ is $104°$. It follows that the sum of the measures of angles $CDA$ and $CAD$ is $\left(180-104\right)°$, or $76°$. Since the measure of angle $CDA$ is equal to the measure of angle $CAD$, the measure of angle $CDA$ is half of $76°$, or $38°$. The sum of the measures of the interior angles of triangle $BDE$ is $180°$. It's given that the measure of angle $EBC$ is $45°$. Since the measure of angle $BDE$, which is the same angle as angle $CDA$, is $38°$, it follows that the measure of angle $DEB$ is $\left(180-45-38\right)°$, or $97°$. Since angle $DEB$ and angle $AEB$ form a straight line, the sum of the measures of these angles is $180°$. It's given in the figure that the measure of angle $AEB$ is $x°$. It follows that $97+x=180$. Subtracting $97$ from both sides of this equation yields $x=83$.

Choice B is correct. It's given that triangles $EFG$ and $JKL$ are congruent such that angle $E$ corresponds to angle $J$. Corresponding angles of congruent triangles are congruent, so angle $E$ and angle $J$ must be congruent. Therefore, if the measure of angle $E$ is $45°$, then the measure of angle $J$ is also $45°$.

Choice A is incorrect. This is the measure of angle $K$, not angle $J$.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $\mathrm{4,205}$. The exterior surface area of a figure is the sum of the areas of all its faces. It's given that the box does not have a lid and that each side of the box is in the shape of a square. Therefore, the box consists of $5$ congruent square faces. It's also given that the length of each edge is $29$ inches. Let $s$ represent the length of an edge of a square. It follows that the area of a square is equal to $s^2$. Therefore, the area of each of the $5$ square faces is equal to ${29}^2$, or $841$, square inches. Since the box consists of $5$ congruent square faces, it follows that the sum of the areas of all its faces, or the exterior surface area of this box without a lid, is $5\left(841\right)$, or $\mathrm{4,205}$, square inches.

The correct answer is $123$. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is $180$ degrees. It's given that the measure of $\angle SQX$ is $48°$ and the measure of $\angle SXQ$ is $86°$. Since points $S$, $Q$, and $X$ form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle QSX$ is $180-48-86$, or $46$. It's also given that the measure of $\angle SWU$ is $85°$. Since $\angle SWU$ and $\angle SWR$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $\angle SWR$ is $180-85$, or $95$. Since points $R$, $S$, and $W$ form a triangle, and $\angle RSW$ is the same angle as $\angle QSX$, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle WRS$ is $180-46-95$, or $39$. It's given that the measure of $\angle VTU$ is $162°$. Since $\angle VTU$ and $\angle STU$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $\angle STU$ is $180-162$, or $18$. Since points $R$, $T$, and $U$ form a triangle, and $\angle URT$ is the same angle as $\angle WRS$, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle TUR$ is $180-39-18$, or $123$.

Choice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle $FGH$ is similar to right triangle $JKL$ and angle $F$ corresponds to angle $J$. It follows that angle $F$ is congruent to angle $J$ and, therefore, the measure of angle $F$ is equal to the measure of angle $J$. The sine ratios of angles of equal measure are equal. Since the measure of angle $F$ is equal to the measure of angle $J$, $\sin \left(F\right)=\sin \left(J\right)$. It's given that $\sin \left(F\right)=\frac{308}{317}$. Therefore,  $\sin \left(J\right)$ is $\frac{308}{317}$.

Choice A is incorrect. This is the value of $\cos \left(J\right)$, not the value of $\sin \left(J\right)$.

Choice C is incorrect. This is the reciprocal of the value of $\sin \left(J\right)$, not the value of $\sin \left(J\right)$.

Choice D is incorrect. This is the reciprocal of the value of $\cos \left(J\right)$, not the value of $\sin \left(J\right)$.

The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of and the other, which is supplementary to the angle with measure , has measure of . Therefore, the value of x is .

Choice D is correct. It's given that lines $m$ and $n$ are parallel. Since line $t$ intersects both lines $m$ and $n$, it's a transversal. The angles in the figure marked as $170°$ and $w°$ are on the same side of the transversal, where one is an interior angle with line $m$ as a side, and the other is an exterior angle with line $n$ as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that $w°=170°$. Therefore, the value of $w$ is $170$. 

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The volume of a right circular cylinder with a radius of r is the product of the area of the base, , and the height, h. The volume of the right circular cylinder described is and its height is 5. If the radius is r, it follows that . Dividing both sides of this equation by yields . Taking the square root of both sides yields or . Since r represents the radius, the value must be positive. Therefore, the radius is 3.

Choice B is incorrect and may result from finding that the square of the radius is 9, but then from dividing 9 by 2, rather than taking the square root of 9. Choice C is incorrect. This represents the square of the radius. Choice D is incorrect and may result from solving the equation for , not r, by dividing by on both sides and then by subtracting, not dividing, 5 from both sides.

The correct answer is $\mathrm{27,556}$. The area of a square is $s^2$, where $s$ is the side length of the square. Let $x$ represent the length of each side of square B. Substituting $x$ for $s$ in $s^2$ yields $x^2$. It follows that the area of square B is $x^2$. It’s given that square A has side lengths that are $166$ times the side lengths of square B. Since $x$ represents the length of each side of square B, the length of each side of square A can be represented by the expression $166x$. It follows that the area of square A is ${\left(166x\right)}^2$, or $\mathrm{27,556}{x}^2$. It’s given that the area of square A is $k$ times the area of square B. Since the area of square A is equal to $\mathrm{27,556}{x}^2$, and the area of square B is equal to $x^2$, an equation representing the given statement is $\mathrm{27,556}{x}^2=k{x}^2$. Since $x$ represents the length of each side of square B, the value of $x$ must be positive. Therefore, the value of $x^2$ is also positive, so it does not equal $0$. Dividing by $x^2$ on both sides of the equation $\mathrm{27,556}{x}^2=k{x}^2$ yields $\mathrm{27,556}=k$. Therefore, the value of $k$ is $\mathrm{27,556}$.

Choice B is correct. The sine of any acute angle is equal to the cosine of its complement. It’s given that in right triangle $RST$, the sum of the measures of angle $R$ and angle $S$ is $90$ degrees. Therefore, angle $R$ and angle $S$ are complementary, and the value of $\sin R$ is equal to the value of $\cos S$. It's given that the value of $\sin R$ is $\frac{\sqrt{15}}{4}$, so the value of $\cos S$ is also $\frac{\sqrt{15}}{4}$.

Choice A is incorrect. This is the value of $\tan S$.

Choice C is incorrect. This is the value of $\frac{1}{\cos S}$.

Choice D is incorrect. This is the value of $\frac{1}{\tan S}$.

Choice C is correct. The area, $A$, of a rectangle is given by the formula $A=\mathcal{l}w$, where $\mathcal{l}$ represents the length of the rectangle and $w$ represents its width. It’s given that the rectangle has a length of $7$ inches and a width of $6$ inches. Substituting $7$ for $\mathcal{l}$ and $6$ for $w$ in the formula $A=\mathcal{l}w$ yields $A=\left(7\right)\left(6\right)$, or $A=42$. Thus, the area, in square inches, of the rectangle is $42$.

Choice A is incorrect. This is the sum, not the product, of the length and width of the rectangle.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is twice the area, in square inches, of the rectangle.

Choice C is correct. The area $A$, $\text{in} \text{square} \text{centimeters} \left({\text{cm}}^2\right)$, of a square with side length $s$, $\text{in} \text{cm}$, is given by the formula $A=s^2$. It’s given that the square has a side length of $55 \text{cm}$. Substituting $55$ for $s$ in the formula $A=s^2$ yields $A={55}^2$, or $A=\mathrm{3,025}$. Therefore, the area, $\text{in} {\text{cm}}^2$, of the square is $\mathrm{3,025}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the perimeter, $\text{in} \text{cm}$, of the square, not its area, $\text{in} {\text{cm}}^2$.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It’s given that triangle R has an area of $80$ ${\mathrm{cm}}^2$. The area of a square is ${\mathcal{l}}^2$, where $\mathcal{l}$ is the side length of the square. It's given that square S has side lengths of $4$ $\mathrm{cm}$. It follows that the area, in ${\mathrm{cm}}^2$, of square S is $4^2$, or $16$. Therefore, the total area, in ${\mathrm{cm}}^2$, of triangle R and square S is $80+16$, or $96$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice C is correct. The area A of a circle with radius r is given by the formula . Thus, a circle with radius 2 has area , which can be rewritten as . The area of a square with side length s is given by the formula . Thus, if a square has the same area as a circle with radius 2, then . Since the side length of a square must be a positive number, taking the square root of both sides of gives . Using the properties of square roots, can be rewritten as , which is equivalent to . Therefore, .

Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression .

The correct answer is $12$. The volume, $V$, of a right square prism can be calculated using the formula $V=s^{2}h$, where $s$ represents the length of an edge of the base and $h$ represents the height of the prism. It’s given that the volume of the prism is $\mathrm{2,016}$ cubic units and the height is $14$ units. Substituting $\mathrm{2,016}$ for $V$ and $14$ for $h$ in the formula $V=s^{2}h$ yields $\mathrm{2,016}=\left(s^2\right)\left(14\right)$. Dividing both sides of this equation by $14$ yields $144=s^2$. Taking the square root of both sides of this equation yields two solutions: $-12=s$ and $12=s$. The length can't be negative, so $12=s$. Therefore, the length, in units, of an edge of the base is $12$.

Choice B is correct. It's given that angle $Z$ in triangle $XYZ$ is a right angle. Thus, side $YZ$ is the leg opposite angle $X$ and side $XZ$ is the leg adjacent to angle $X$. The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. It follows that $\tan X=\frac{YZ}{XZ}$. It's given that $\tan X=\frac{12}{35}$ and the length of side $YZ$ is $24$ units. Substituting $\frac{12}{35}$ for $\tan X$ and $24$ for $YZ$ in the equation $\tan X=\frac{YZ}{XZ}$ yields $\frac{12}{35}=\frac{24}{XZ}$. Multiplying both sides of this equation by $35\left(XZ\right)$ yields $12\left(XZ\right)=24\left(35\right)$, or $12\left(XZ\right)=840$. Dividing both sides of this equation by $12$ yields $XZ=70$. The length $XY$ can be calculated using the Pythagorean theorem, which states that if a right triangle has legs with lengths of $a$ and $b$ and a hypotenuse with length $c$, then $a^2+b^2=c^2$. Substituting $70$ for $a$ and $24$ for $b$ in this equation yields ${70}^2+{24}^2=c^2$, or $\mathrm{5,476}=c^2$. Taking the square root of both sides of this equation yields $\pm 74=c$. Since the length of the hypotenuse must be positive, $74=c$. Therefore, the length of $XY$ is $74$ units. The perimeter of a triangle is the sum of the lengths of all sides. Thus, $\left(74+70+24\right)$ units, or $168$ units, is the perimeter of triangle $XYZ$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This would be the perimeter, in units, for a right triangle where the length of side $YZ$ is $12$ units, not $24$ units.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The sine of an angle is equal to the cosine of its complementary angle. In the triangle shown, angle $Z$ is a right angle; thus, angles $X$ and $Y$ are complementary angles. Therefore, $\cos Y$ has the same value as $\sin X$.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice B is correct. Since , the measures of their corresponding angles, and , are equal. Since has a measure of , the measure of is also . Since the sum of the measures of the interior angles in a triangle is , it follows that , or . Subtracting by 134 on both sides of this equation yields .

Choices A, C, and D are incorrect and may result from calculation errors.

The correct answer is 30. It is given that the measure of is . Angle MPR and are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is , and so the measure of is . The sum of the angles in a triangle is . Subtracting the measure of from yields the sum of the other angles in the triangle MPR. Since , the sum of the measures of and is . It is given that , so it follows that triangle MPR is isosceles. Therefore and must be congruent. Since the sum of the measure of these two angles is , it follows that the measure of each angle is .

An alternate approach would be to use the exterior angle theorem, noting that the measure of is equal to the sum of the measures of and . Since both angles are equal, each of them has a measure of .

Choice D is correct. The perimeter, $P$, of a square can be found using the formula $P=4s$, where $s$ is the length of each side of the square. It's given that square X has a side length of $12$ centimeters. Substituting $12$ for $s$ in the formula for the perimeter of a square yields $P=4\left(12\right)$, or $P=48$. Therefore, the perimeter of square X is $48$ centimeters. It’s also given that the perimeter of square Y is $2$ times the perimeter of square X. Therefore, the perimeter of square Y is $2\left(48\right)$, or $96$, centimeters. Substituting $96$ for $P$ in the formula $P=4s$ gives $96=4s$. Dividing both sides of this equation by $4$ gives $24=s$. Therefore, the length of one side of square Y is $24$ centimeters.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice C is correct. The volume, $V$, of a right circular cylinder is given by the formula $V=\pi r^{2}h$, where $r$ is the radius of the base of the cylinder and $h$ is the height of the cylinder. It’s given that a right circular cylinder has a height of $6$ meters. Therefore, $h=6$. It's also given that the right circular cylinder has a base with a circumference of $20\pi$ meters. The circumference, $C$, of a circle is given by $C=2\pi r$, where $r$ is the radius of the circle. Substituting $20\pi$ for $C$ in the formula $C=2\pi r$ yields $20\pi =2\pi r$. Dividing each side of this equation by $2\pi$ yields $10=r$. Substituting $10$ for $r$ and $6$ for $h$ in the formula $V=\pi r^{2}h$ yields $V=\pi {\left(10\right)}^2\left(6\right)$, or $V=600\pi$. Therefore, the volume, in cubic meters, of the cylinder is $600\pi$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the lateral surface area, not the volume, of the cylinder.

Choice D is incorrect. This is the result of using the diameter, not the radius, for the value of $r$ in the formula $V=\pi r^{2}h$.

Choice B is correct. In similar triangles, corresponding angles are congruent. It’s given that right triangles $PQR$ and $STU$ are similar, where angle $P$ corresponds to angle $S$. It follows that angle $P$ is congruent to angle $S$. In the triangles shown, angle $R$ and angle $U$ are both marked as right angles, so angle $R$ and angle $U$ are corresponding angles. It follows that angle $Q$ and angle $T$ are corresponding angles, and thus, angle $Q$ is congruent to angle $T$. It’s given that the measure of angle $Q$ is $18°$, so the measure of angle $T$ is also $18°$. Angle $U$ is a right angle, so the measure of angle $U$ is $90°$. The sum of the measures of the interior angles of a triangle is $180°$. Thus, the sum of the measures of the interior angles of triangle $STU$ is $180$ degrees. Let $s$ represent the measure, in degrees, of angle $S$. It follows that $s+18+90=180$, or $s+108=180$. Subtracting $108$ from both sides of this equation yields $s=72$. Therefore, if the measure of angle $Q$ is $18$ degrees, then the measure of angle $S$ is $72$ degrees.

Choice A is incorrect. This is the measure of angle $T$.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the sum of the measures of angle $S$ and angle $U$.

Choice A is correct. It’s given that triangle $JKL$ is similar to triangle $RST$, where $J$ corresponds to $R$ and $K$ corresponds to $S$. It follows that $\overline{JK}$ corresponds to $\overline{RS}$. If two triangles are similar, then the scale factor between their perimeters is equal to the scale factor between the lengths of their corresponding sides. It's given that the length of $\overline{JK}$ is $15$ and the length of $\overline{RS}$ is $135$. Therefore, the scale factor from the length of $\overline{JK}$ to the length of $\overline{RS}$ is $\frac{135}{15}$, or $9$. It’s given that the perimeter of triangle $JKL$ is $36$. Let $p$ represent the perimeter of triangle $RST$. It follows that $\frac{p}{36}=9$. Multiplying each side of this equation by $36$ yields $p=324$. Therefore, the perimeter of triangle $RST$ is $324$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $57$. Based on the figure, the angle with measure $y°$ and the angle vertical to the angle with measure $58°$ are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure $58°$ also has measure $58°$. It’s given that lines $q$ and $r$ are parallel. Therefore, same side interior angles between lines $q$ and $r$ are supplementary. It follows that $y+58=180$. If $y=2x+8$, then the value of $x$ can be found by substituting $2x+8$ for $y$ in the equation $y+58=180$, which yields $\left(2x+8\right)+58=180$, or $2x+66=180$. Subtracting $66$ from both sides of this equation yields $2x=114$. Dividing both sides of this equation by $2$ yields $x=57$. Thus, if $y=2x+8$, the value of $x$ is $57$.

Choice B is correct. Let the measure of the third angle in the smaller triangle be . Since lines and m are parallel and cut by transversals, it follows that the corresponding angles formed are congruent. So . The sum of the measures of the interior angles of a triangle is , which for the interior angles in the smaller triangle yields . Given that and , it follows that . Solving for x gives , or .

Choice A is incorrect and may result from incorrectly assuming that angles . Choice C is incorrect and may result from incorrectly assuming that the smaller triangle is a right triangle, with x as the right angle. Choice D is incorrect and may result from a misunderstanding of the exterior angle theorem and incorrectly assuming that .

Choice B is correct. Since , , and are parallel, quadrilaterals and are similar. Let x represent the length of . With similar figures, the ratios of the lengths of corresponding sides are equal. It follows that . Multiplying both sides of this equation by 18.5 and by x yields , or . Dividing both sides of this equation by 9 yields , which to the nearest tenth is 17.5.

Choices A, C, and D are incorrect and may result from errors made when setting up the proportion.

Choice A is correct. It's given that the area of a square is $64$ square inches. The area $A$, in square inches, of a square is given by the formula $A=s^2$, where $s$ is the side length, in inches, of the square. Substituting $64$ for $A$ in this formula yields $64=s^2$. Taking the positive square root of both sides of this equation yields $8=s$. Thus, the side length, in inches, of this square is $8$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the area, in square inches, of the square, not the side length, in inches, of the square.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. Angle A is an acute angle in a right triangle, so the value of tan(A) is equivalent to the ratio of the length of the side opposite angle A, 20, to the length of the nonhypotenuse side adjacent to angle A, 21. Therefore,  .

Choice A is incorrect. This is the value of sin(A). Choice B is incorrect. This is the value of cos(A). Choice D is incorrect. This is the value of tan(B).

Choice D is correct. The area, $A$, of a circle is given by the formula $A=\pi r^2$, where $r$ is the radius of the circle. It’s given that the circle has a radius of $43$ meters. Substituting $43$ for $r$ in the formula $A=\pi r^2$ yields $A=\pi {\left(43\right)}^2$, or $A=\mathrm{1,849}\pi$. Therefore, the area, in square meters, of the circle is $\mathrm{1,849}\pi$.

Choice A is incorrect. This is the area, in square meters, of a circle with a radius of $\sqrt{\frac{43}{2}}$ meters.

Choice B is incorrect. This is the area, in square meters, of a circle with a radius of $\sqrt{43}$ meters.

Choice C is incorrect. This is the circumference, in meters, of the circle.

Choice C is correct. It’s given that the circle has its center at $\left(-1,1\right)$ and that line $t$ is tangent to this circle at the point $\left(5,-4\right)$. Therefore, the points $\left(-1,1\right)$ and $\left(5,-4\right)$ are the endpoints of the radius of the circle at the point of tangency. The slope of a line or line segment that contains the points $\left(a,b\right)$ and $\left(c,d\right)$ can be calculated as $\frac{d-b}{c-a}$. Substituting $\left(-1,1\right)$ for $\left(a,b\right)$ and $\left(5,-4\right)$ for $\left(c,d\right)$ in the expression $\frac{d-b}{c-a}$ yields $\frac{-4-1}{5-\left(-1\right)}$ , or $-\frac{5}{6}$. Thus, the slope of this radius is $-\frac{5}{6}$. A line that’s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line $t$ is perpendicular to the radius at the point $\left(5,-4\right)$, so the slope of line $t$ is the negative reciprocal of the slope of this radius. The negative reciprocal of $-\frac{5}{6}$ is $\frac{6}{5}$. Therefore, the slope of line $t$ is $\frac{6}{5}$. Since the slope of line $t$ is the same between any two points on line $t$, a point lies on line $t$ if the slope of the line segment connecting the point and $\left(5,-4\right)$ is $\frac{6}{5}$. Substituting choice C, $\left(10,2\right)$, for $\left(a,b\right)$ and $\left(5,-4\right)$ for $\left(c,d\right)$ in the expression $\frac{d-b}{c-a}$ yields $\frac{-4-2}{5-10}$, or $\frac{6}{5}$. Therefore, the point $\left(10,2\right)$ lies on line $t$.

Choice A is incorrect. The slope of the line segment connecting $\left(0,\frac{6}{5}\right)$ and $\left(5,-4\right)$ is $\frac{-4-{\displaystyle \frac{6}{5}}}{5-0}$, or $-\frac{26}{25}$, not $\frac{6}{5}$.

Choice B is incorrect. The slope of the line segment connecting $\left(4,7\right)$ and $\left(5,-4\right)$ is $\frac{-4-7}{5-4}$, or $-11$, not $\frac{6}{5}$.

Choice D is incorrect. The slope of the line segment connecting $\left(11,1\right)$ and $\left(5,-4\right)$ is $\frac{-4-1}{5-11}$, or $\frac{5}{6}$, not $\frac{6}{5}$.

Choice B is correct. Since it's given that $P$ is the center of a circle with a radius of $9$ inches, and that points $Q$ and $R$ lie on that circle, it follows that $\overline{PQ}$ and $\overline{RP}$ of triangle $PQR$ each have a length of $9$ inches. Let the length of $\overline{QR}$ be $x$ inches. It follows that the perimeter of triangle $PQR$ is $9+9+x$ inches. Since it's given that the perimeter of triangle $PQR$ is $31$ inches, it follows that $9+9+x=31$, or $18+x=31$. Subtracting $18$ from both sides of this equation gives $x=13$. Therefore, the length, in inches, of $\overline{QR}$ is $13$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. It’s given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point $(7,8)$ is halfway between the points $(7,5)$ and $(7,11)$ and is a distance of $3$ units from each of those two points. The point $(7,8)$ is also a distance of $3$ units from $(4,8)$. Because the point $(7,8)$ is the same distance from all three given points, it must be the center of the circle. The radius of a circle is the distance from the center to any point on the circle. Since that distance is $3$, it follows that the radius of the circle is $3$. The circumference of a circle with radius $r$ is equal to $2\pi r$. It follows that the circumference of the circle is $2\left(\pi \right)\left(3\right)$, or $6\pi$. It's given that the circumference of the circle is $k\pi$. Therefore, the value of $k$ is $6$.

Choice A is incorrect. This is the radius of the circle, not the value of $k$ in the expression $k\pi$.

Choice C is incorrect. This is the x-coordinate of the center of the circle, not the value of $k$ in the expression $k\pi$.

Choice D is incorrect. This is the value of $k$ for which $k\pi$ represents the area of the circle, in square units, not the circumference of the circle, in units.

The correct answer is 360. The volume of a right rectangular prism is calculated by multiplying its dimensions: length, width, and height. Multiplying the values given for these dimensions yields a volume of cubic centimeters.

Choice A is correct. The volume of a cube can be found by using the formula $V=s^3$, where $V$ is the volume and $s$ is the edge length of the cube. Therefore, the volume of the given cube is $V={68}^3$, or $\mathrm{314,432}$ cubic inches. The volume of a sphere can be found by using the formula $V=\frac{4}{3}\pi r^3$ , where $V$ is the volume and $r$ is the radius of the sphere. Therefore, the volume of the given sphere is $V=\frac{4}{3}\pi {\left(34\right)}^3$, or approximately $\mathrm{164,636}$ cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives $\mathrm{314,432}-\mathrm{164,636}=\mathrm{149,796}$ cubic inches.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $880$. The volume, $V$, of a right rectangular prism is given by the formula $V=\mathcal{l}wh$, where $\mathcal{l}$ is the length, $w$ is the width, and $h$ is the height of the prism. It’s given that a right rectangular prism has a length of $11$ meters, a width of $8$ meters, and a height of $10$ meters. Substituting $11$ for $\mathcal{l}$, $8$ for $w$, and $10$ for $h$ in the formula $V=\mathcal{l}wh$ yields $V=\left(11\right)\left(8\right)\left(10\right)$, or $V=880$. Therefore, the volume, in cubic meters, of the prism is $880$.

Choice D is correct. It's given that in triangle $ABC$, the measure of angle $B$ is $90°$ and  $\overline{BD}$ is an altitude of the triangle. Therefore, the measure of angle $BDC$ is $90°$. It follows that angle $B$ is congruent to angle $D$ and angle $C$ is congruent to angle $C$. By the angle-angle similarity postulate, triangle $ABC$ is similar to triangle $BDC$. Since triangles $ABC$ and $BDC$ are similar, it follows that $\frac{AC}{AB}=\frac{BC}{BD}$. It's also given that the length of $\overline{AB}$ is $15$ and the length of $\overline{AC}$ is $23$ greater than the length of $\overline{AB}$. Therefore, the length of $\overline{AC}$ is $15+23$, or $38$. Substituting $15$ for $AB$ and $38$ for $AC$ in the equation $\frac{AC}{AB}=\frac{BC}{BD}$ yields $\frac{38}{15}=\frac{BC}{BD}$. Therefore, the value of $\frac{BC}{BD}$ is $\frac{38}{15}$.

Choice A is incorrect. This is the value of $\frac{BD}{BC}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.
 

Choice C is correct. The volume of a right circular cylinder is equal to $\pi a^{2}b$, where $a$ is the radius of a base of the cylinder and $b$ is the height of the cylinder. It’s given that the cylinder shown has a radius of $r$ and a height of $h$. It follows that the volume of the cylinder shown is equal to $\pi r^{2}h$. It’s given that the second right circular cylinder has a radius of $R$ and a height of $H$. It follows that the volume of the second cylinder is equal to $\pi R^{2}H$. Choice C gives $R=7r$ and $H=8h$. Substituting $7r$ for $R$ and $8h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(7r\right)}^2\left(8h\right)$, or $\pi \left(49r^2\right)\left(8h\right)$, which is equivalent to $\pi \left(392r^{2}h\right)$, or $392\left(\pi r^{2}h\right)$. This expression is equal to $392$ times the volume of the cylinder shown, $\pi r^{2}h$. Therefore, $R=7r$ and $H=8h$ could represent the radius $R$, in terms of $r$, and the height $H$, in terms of $h$, of the second cylinder.

Choice A is incorrect. Substituting $8r$ for $R$ and $7h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(8r\right)}^2\left(7h\right)$, or $\pi \left(64r^2\right)\left(7h\right)$, which is equivalent to $\pi \left(448r^{2}h\right)$, or $448\left(\pi r^{2}h\right)$. This expression is equal to $448$, not $392$, times the volume of the cylinder shown. 

Choice B is incorrect. Substituting $8r$ for $R$ and $49h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(8r\right)}^2\left(49h\right)$, or $\pi \left(64r^2\right)\left(49h\right)$, which is equivalent to $\pi \left(\mathrm{3,136}{r}^{2}h\right)$, or $\mathrm{3,136}\left(\pi r^{2}h\right)$. This expression is equal to $\mathrm{3,136}$, not $392$, times the volume of the cylinder shown.

Choice D is incorrect. Substituting $49r$ for $R$ and $8h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(49r\right)}^2\left(8h\right)$, or $\pi \left(\mathrm{2,401}{r}^2\right)\left(8h\right)$, which is equivalent to $\pi \left(\mathrm{19,208}{r}^{2}h\right)$, or $\mathrm{19,208}\left(\pi r^{2}h\right)$. This expression is equal to $\mathrm{19,208}$, not $392$, times the volume of the cylinder shown.

Choice A is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is $180°$. It's given that in $△XYZ$, the measure of $\angle X$ is $24°$ and the measure of $\angle Y$ is $98°$. It follows that the measure of $\angle Z$ is $\left(180-24-98\right)°$, or $58°$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the sum of the measures of $\angle X$ and $\angle Y$, not the measure of $\angle Z$.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $29$. The volume, $V$, of a right circular cylinder is given by the formula $V=\pi r^{2}h$, where $r$ is the radius of the base of the cylinder and $h$ is the height of the cylinder. Since the base of the cylinder is a circle with radius $r$, the area of the base of the cylinder is $\pi r^2$. It's given that a right circular cylinder has a volume of $377$ cubic centimeters; therefore, $V=377$. It's also given that the area of the base of the cylinder is $13$ square centimeters; therefore, $\pi r^2=13$. Substituting $377$ for $V$ and $13$ for $\pi r^2$ in the formula $V=\pi r^{2}h$ yields $377=13h$. Dividing both sides of this equation by $13$ yields $29=h$. Therefore, the height of the cylinder, in centimeters, is $29$.

Choice B is correct. Each tree and its shadow can be modeled using a right triangle, where the height of the tree and the length of its shadow are the legs of the triangle. At a given point in time, the right triangles formed by two nearby trees and their respective shadows will be similar. Therefore, if the height of the other tree is $x$, in feet, the value of $x$ can be calculated by solving the proportional relationship $\frac{10 \text{feet tall}}{5 \text{feet long}}=\frac{x \text{feet tall}}{2 \text{feet long}}$. This equation is equivalent to$\frac{10}{5}=\frac{x}{2}$, or $2=\frac{x}{2}$. Multiplying each side of the equation $2=\frac{x}{2}$ by $2$ yields $4=x$. Therefore, the other tree is $4 \text{feet}$ tall.

Choice A is incorrect and may result from calculating the difference between the lengths of the shadows, rather than the height of the other tree.

Choice C is incorrect and may result from calculating the difference between the height of the $10$-foot-tall tree and the length of the shadow of the other tree, rather than calculating the height of the other tree.

Choice D is incorrect and may result from a conceptual or calculation error.

Choice A is correct. The figure shows that angle $P$ in $△QPR$ and angle $T$ in $△STR$ are right angles. It follows that angle $P$ is congruent to angle $T$. The figure also shows that the measures of angle $QRP$ and angle $SRT$ are both $x°$. Therefore, angle $QRP$ is congruent to angle $SRT$. It’s given that $△QPR$ is similar to $△STR$. Since angle $P$ is congruent to angle $T$, and angle $QRP$ is congruent to angle $SRT$, it follows that $\overline{QR}$ corresponds to $\overline{SR}$, and $\overline{QP}$ corresponds to $\overline{ST}$. Since corresponding sides of similar triangles are proportional, it follows that $\frac{SR}{QR}=\frac{ST}{QP}$. It’s also given that the lengths of $\overline{ST}$, $\overline{QP}$, and $\overline{QR}$ are $14$, $15$, and $25$, respectively. Substituting $14$ for $ST$, $15$ for $QP$, and $25$ for $QR$ in the equation $\frac{SR}{QR}=\frac{ST}{QP}$ yields $\frac{SR}{25}=\frac{14}{15}$. Multiplying each side of this equation by $25$ yields $SR=\left(\frac{14}{15}\right)\left(25\right)$, or $SR=\frac{350}{15}$. Thus, the length of $\overline{SR}$ is $\frac{350}{15}$.

Choice B is incorrect. This is the result of solving the equation $\frac{SR}{25}=\frac{14}{20}$, not $\frac{SR}{25}=\frac{14}{15}$.

Choice C is incorrect. This is the result of solving the equation $\frac{SR}{14}=\frac{15}{20}$, not $\frac{SR}{25}=\frac{14}{15}$.

Choice D is incorrect. This is the result of solving the equation $\frac{SR}{14}=\frac{15}{25}$, not $\frac{SR}{25}=\frac{14}{15}$.

Choice D is correct. It's given that lines $m$ and $n$ are parallel. Since line $t$ intersects both lines $m$ and $n$, it's a transversal. The angles in the figure marked as $134°$ and $w°$ are on the same side of the transversal, where one is an interior angle with line $m$ as a side, and the other is an exterior angle with line $n$ as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that $w°=134°$. Therefore, the value of $w$ is $134$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It's given that triangle $ABC$ is similar to triangle $XYZ$, where $A$, $B$, and $C$ correspond to $X$, $Y$, and $Z$, respectively. It follows that side $AB$ corresponds to side $XY$ and side $BC$ corresponds to side $YZ$. Since the lengths of corresponding sides in similar triangles are proportional, it follows that $\frac{XY}{AB}=\frac{YZ}{BC}$. Substituting $170$ for $AB$, $60$ for $YZ$, and $850$ for $BC$ in this equation yields $\frac{XY}{170}=\frac{60}{850}$. Multiplying each side of this equation by $170$ yields $XY=12$. Therefore, the length of $\overline{XY}$ is $12$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the length of $\overline{YZ}$, not $\overline{XY}$.

The correct answer is $156$. In the figure shown, the sum of the measures of angle $UVS$ and angle $RVS$ is $180°$. It’s given that the measure of angle $RVS$ is $41°$. Therefore, the measure of angle $UVS$ is $\left(180-41\right)°$, or $139°$. The sum of the measures of the interior angles of a triangle is $180°$. In triangle $UVS$, the measure of angle $UVS$ is $139°$ and it's given that the measure of angle $VST$ is $29°$. Thus, the measure of angle $VUS$ is $\left(180-139-29\right)°$, or $12°$. It’s given that $RT=TU$. Therefore, triangle $TUR$ is an isosceles triangle and the measure of $VUS$ is equal to the measure of angle $TRU$. In triangle $TUR$, the measure of angle $VUS$ is $12°$ and the measure of angle $TRU$ is $12°$. Thus, the measure of angle $UTR$ is $\left(180-12-12\right)°$, or $156°$. The figure shows that the measure of angle $UTR$ is $x°$, so the value of $x$ is $156$.

The correct answer is $54$. The figure shown includes two triangles, triangle $WQX$ and triangle $YQZ$, such that angle $WQX$ and angle $YQZ$ are vertical angles. It follows that angle $WQX$ is congruent to angle $YQZ$. It’s also given in the figure that the measures of angle $W$ and angle $Y$ are $a°$. Therefore angle $W$ is congruent to angle $Y$. Since triangle $WQX$ and triangle $YQZ$ have two pairs of congruent angles, triangle $WQX$ is similar to triangle $YQZ$ by the angle-angle similarity postulate, where $\overline{YZ}$ corresponds to $\overline{WX}$, and $\overline{YQ}$ corresponds to $\overline{WQ}$. Since the lengths of corresponding sides in similar triangles are proportional, it follows that $\frac{YZ}{WX}=\frac{YQ}{WQ}$. It’s given that $YQ=63$, $WQ=70$, and $WX=60$. Substituting $63$ for $YQ$, $70$ for $WQ$, and $60$ for $WX$ in the equation $\frac{YZ}{WX}=\frac{YQ}{WQ}$ yields $\frac{YZ}{60}=\frac{63}{70}$. Multiplying each side of this equation by $60$ yields $YZ=\left(\frac{63}{70}\right)\left(60\right)$, or $YZ=54$. Therefore, the length of $\overline{YZ}$ is $54$.

Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc is subtended by a central angle with measure . If s is the length of arc , then s must satisfy the ratio . Reducing the fraction to its simplest form gives . Therefore, . Multiplying both sides of by yields .

Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc , not its full length.

Choice C is correct. Since angles $B$ and $E$ each have the same measure and angles $C$ and $F$ each have the same measure, triangles $ABC$ and $DEF$ are similar, where side $BC$ corresponds to side $EF$. To determine whether two similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent. Therefore, to determine whether triangles $ABC$ and $DEF$ are congruent, it is sufficient to determine whether sides $BC$ and $EF$ have equal length. Thus, the lengths of $BC$ and $EF$ are sufficient to determine whether triangle $ABC$ is congruent to triangle $DEF$.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect. The given information is sufficient to determine that triangles $ABC$ and $DEF$ are similar, but not whether they are congruent.

Choice B is correct. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to angle $A$ is $41$, and the length of the hypotenuse is $42$. Therefore, $\cos A=\frac{41}{42}$.

Choice A is incorrect. This is the value of $\frac{1}{\cos A}$.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is $338$. The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In triangle $XYZ$, it's given that angle $Y$ is a right angle. Thus, $\overline{XY}$ is the leg opposite of angle $Z$ and $\overline{YZ}$ is the leg adjacent to angle $Z$. It follows that $\tan Z=\frac{XY}{YZ}$. It's also given that the measure of angle $Z$ is $33°$ and the length of $\overline{YZ}$ is $26$ units. Substituting $33°$ for $Z$ and $26$ for $YZ$ in the equation $\tan Z=\frac{XY}{YZ}$ yields $\tan 33°=\frac{XY}{26}$. Multiplying each side of this equation by $26$ yields $26\tan 33°=XY$. Therefore, the length of $\overline{XY}$ is $26\tan 33°$. The area of a triangle is half the product of the lengths of its legs. Since the length of $\overline{YZ}$ is $26$ and the length of $\overline{XY}$ is $26\tan 33°$, it follows that the area of triangle $XYZ$ is $\frac{1}{2}\left(26\right)\left(26\tan 33°\right)$ square units, or $338\tan 33°$ square units. It's given that the area, in square units, of triangle $XYZ$ can be represented by the expression $k\tan 33°$, where $k$ is a constant. Therefore, $338$ is the value of $k$.

The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle B, or , is equal to the ratio of the length of side to the length of side . It’s given that the length of side is 26 and that . Therefore, . Multiplying both sides of this equation by 26 yields .

By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle ABC is as follows: , or . Subtracting 100 from both sides of yields . Taking the square root of both sides of yields .